APPROXIMATE AND LOW REGULARITY DIRICHLET BOUNDARY CONDITIONS IN THE GENERALIZED FINITE ELEMENT METHOD IVO BABU ˇ SKA, VICTOR NISTOR, AND NICOLAE TARFULEA Abstract. We propose a method for treating the Dirichlet boundary condi- tions in the framework of the Generalized Finite Element Method (GFEM). We are especially interested in boundary data with low regularity (possibly a distribution). We use approximate Dirichlet boundary conditions as in [11] and polynomial approximations of the boundary. Our sequence of GFEM- spaces considered, Sμ, μ =1, 2,... is such that Sμ ⊂ H 1 0 (Ω), and hence it does not conform to one of the basic FEM conditions. Let hμ be the typ- ical size of the elements defining Sμ and let u ∈ H m+1 (Ω) be the solution of the Poisson problem -Δu = f in Ω, u = 0 on ∂Ω, on a smooth, bounded domain Ω. Assume that vμH 1/2 (∂Ω) ≤ Ch m μ vμH 1 (Ω) for all vμ ∈ Sμ and |u - u I | H 1 (Ω) ≤ Ch m μ uH m+1 (Ω) , u ∈ H m+1 (Ω) ∩ H 1 0 (Ω), for a suit- able u I ∈ Sμ. Then we prove that we obtain quasi-optimal rates of con- vergence for the sequence uμ ∈ Sμ of GFEM approximations of u, that is, u - uμH 1 (Ω) ≤ Ch m μ f H m-1 (Ω) . We also extend our results to the inho- mogeneous Dirichlet boundary value problem -Δu = f in Ω, u = g on ∂Ω, including the case when f = 0 and g has low regularity (i.e., is a distribu- tion). Finally, we indicate an effective technique for constructing sequences of GFEM-spaces satisfying our conditions using polynomial approximations of the boundary. Contents Introduction 2 Part 1. Approximate Dirichlet boundary conditions 5 1. Homogeneous boundary conditions 5 2. Non-homogeneous boundary conditions 8 3. Distributional boundary data and the “inf-sup” condition 9 Part 2. GFEM Approximation Spaces 11 4. The Generalized Finite Element Method 11 5. Properties of the spaces S μ 15 6. Interior numerical approximation 21 7. Comments and further problems 24 References 24 Date : April 30, 2007. I. Babuˇ ska was partially supported by NSF Grant DMS 0341982 and ONR Grant N00014-94- 0401. V. Nistor was partially supported by NSF Grant DMS 0555831. 1
Transcript
APPROXIMATE AND LOW REGULARITY DIRICHLETBOUNDARY CONDITIONS IN THE GENERALIZED FINITE
ELEMENT METHOD
IVO BABUSKA, VICTOR NISTOR, AND NICOLAE TARFULEA
Abstract. We propose a method for treating the Dirichlet boundary condi-tions in the framework of the Generalized Finite Element Method (GFEM).
We are especially interested in boundary data with low regularity (possibly
a distribution). We use approximate Dirichlet boundary conditions as in [11]and polynomial approximations of the boundary. Our sequence of GFEM-
spaces considered, Sµ, µ = 1, 2, . . . is such that Sµ 6⊂ H10 (Ω), and hence it
does not conform to one of the basic FEM conditions. Let hµ be the typ-ical size of the elements defining Sµ and let u ∈ Hm+1(Ω) be the solution
of the Poisson problem −∆u = f in Ω, u = 0 on ∂Ω, on a smooth, bounded
domain Ω. Assume that ‖vµ‖H1/2(∂Ω) ≤ Chmµ ‖vµ‖H1(Ω) for all vµ ∈ Sµ
and |u − uI |H1(Ω) ≤ Chmµ ‖u‖Hm+1(Ω), u ∈ Hm+1(Ω) ∩ H1
0 (Ω), for a suit-
able uI ∈ Sµ. Then we prove that we obtain quasi-optimal rates of con-vergence for the sequence uµ ∈ Sµ of GFEM approximations of u, that is,
‖u − uµ‖H1(Ω) ≤ Chmµ ‖f‖Hm−1(Ω). We also extend our results to the inho-
mogeneous Dirichlet boundary value problem −∆u = f in Ω, u = g on ∂Ω,
including the case when f = 0 and g has low regularity (i.e., is a distribu-tion). Finally, we indicate an effective technique for constructing sequences
of GFEM-spaces satisfying our conditions using polynomial approximations ofthe boundary.
Contents
Introduction 2
Part 1. Approximate Dirichlet boundary conditions 51. Homogeneous boundary conditions 52. Non-homogeneous boundary conditions 83. Distributional boundary data and the “inf-sup” condition 9
Part 2. GFEM Approximation Spaces 114. The Generalized Finite Element Method 115. Properties of the spaces Sµ 156. Interior numerical approximation 217. Comments and further problems 24References 24
Date: April 30, 2007.I. Babuska was partially supported by NSF Grant DMS 0341982 and ONR Grant N00014-94-
0401. V. Nistor was partially supported by NSF Grant DMS 0555831.
1
2 I BABUSKA, V. NISTOR, AND N. TARFULEA
Introduction
In the past few years, meshless methods for the approximation of solutions ofpartial differential equations have received increasing attention, especially in theEngineering and Physics communities. The reasons behind the development ofsuch methods are the difficulties associated to the mesh generation, particularlywhen the geometry of the domain is complicated. As in the case of the usual FiniteElement Method, one of the major problems in the implementation of meshlessmethods is the enforcement of Dirichlet boundary conditions. It is the purpose ofthis paper to address the problem of enforcing Dirichlet boundary conditions in theGeneralized Finite Element Method framework. We are especially interested in thecase when the Dirichlet boundary data has low regularity, including the case whenit is a distribution on the boundary.
The classical Rayleigh-Ritz method for elliptic Dirichlet boundary value prob-lems requires that the trial subspace functions fulfill the boundary conditions. Nev-ertheless, the construction of such subspaces implies many difficulties in practicewhen the boundary of the domain is curved. Therefore, several approaches havebeen devised for dealing with the Dirichlet boundary conditions on domains withcurved boundaries. One approach is to modify the variational principles by addingappropriate boundary terms so that there will be no need for the trial subspaces tofulfill any condition at the boundary. See the works of Babuska [2, 4], Bramble andNitsche [12], and Bramble and Schatz [13, 14], among others, for examples of howthis approach works in practice. Another approach (used also in this paper) is touse subspaces with nearly zero boundary conditions. This ideea was first outlinedby Nitsche [29] and further studied by Berger, Scott, and Strang [11] and Nitsche[30].
Yet another approach for dealing with the Dirichlet boundary conditions is theIsoparametric Finite Element Method or IFEM with curved finite elements alongthe boundary. See [18] and references therein, or [17, 19, 21, 23, 24, 35, 36], amongmany others, for more recent work and applications. This approach is typicallyused in connection with a numerical quadrature scheme computing the coefficientsof the resulting linear systems. In the applications of this method, except in specialcases (such as when Ω is a polyhedral domain) the interior Ωh of the union of thefinite elements is not equal to Ω, although the boundary of Ωh is very close to ∂Ω.That is, the approximate solution uh is sought in a subspace Vh ⊂ H1
0 (Ωh), and sothe homogeneous Dirichlet boundary condition u = 0 on ∂Ω is “approximated” bythe boundary condition uh = 0 on ∂Ωh. In fact, uh is the solution of a variationalequation ah(uh, vh) = (fh, vh)h for all vh ∈ Vh, where ah(·, ·) is a bilinear form whichapproximates the usual bilinear form defined over H1(Ωh)×H1(Ωh), and fh ∈ V ∗
h
approximates the linear form vh ∈ Vh →∫Ωhfvhdx, where f is an extension of f
to the set Ωh.Our approach has certain points in common with the isoparametric method just
mentioned in the fact that we are using polynomial approximations of the boundary.However, our method does not require non-linear changes of coordinates. Ourmethod thus combines the approaches in the papers of Berger, Scott, and Strang[11] and Nitsche [30]. Our definition of the discrete solution is as in [11], whereas ourassumptions are closer to those of [30]. We have tried to keep our assumptions at aminimum. This is possible using partitions of unity, more precisely the Generalized
DIRICHLET PROBLEM 3
Finite Element Method or GFEM, a method that originated in the work of Babuska,Caloz, and Osborn [6] and further developed in [4, 7, 8, 10, 22, 25, 27, 37].
Our construction is different from the IFEM in that we do not require compli-cated non-linear changes of coordinates. Moreover, our method uses non-conformingsubspaces of functions and it does not have to deal with extensions over larger do-mains. It is closely related to [9] which uses GFEM for elliptic Neumann boundaryvalue problems with distributional boundary data. The GFEM is a generalizationof the meshless methods which use the idea of partition of unity. This methodallows a great flexibility in constructing the trial spaces, permits inclusion of apriori knowledge about the differential equation in the trial spaces, and gives theoption of constructing trial spaces of any desired regularity. We mention that theGFEM is also known and used under other names, such as: the method of “clouds,”the method of “finite spheres,” the “X–finite element method,” and others. See thesurvey by Babuska, Banerjee, and Osborn [4] for further information and references.
Let us now describe the main results of this paper in some detail. Let Ω ⊂ Rn bea smooth, bounded domain with boundary ∂Ω. Here are some of our assumptions.Let f ∈ L2(Ω) and u ∈ H2(Ω) be the unique solution of the Poisson problem
(1) −∆u = f on Ω, u = 0 on ∂Ω.
Assume that we are given a sequence hµ → 0 and a sequence Sµ ⊂ H1(Ω) of test-trial spaces. The parameters hµ play the role of the size of the elements definingSµ. Let B(u, v) :=
∫Ω∇u · ∇vdx. We define the discrete solution uµ ∈ Sµ in the
usual way:
(2) B(uµ, vµ) = 〈f, vµ〉L2(Ω) :=∫
Ω
fvµ dx, for all vµ ∈ Sµ.
We do not assume however that Sµ satisfy exactly the Dirichlet boundary condi-tions, that is, we do not assume Sµ ⊂ H1
0 (Ω).Let us fix from now on a natural number m ∈ N = 1, 2, . . . that will play, in
what follows, the role of the expected order of approximation. We shall make thefollowing two basic assumptions. The first assumption is that our approximatingfunctions satisfy Dirichlet boundary conditions approximately:
for any vµ ∈ Sµ, µ ∈ N.The second assumption is an approximation assumption that will be used also for
non-homogeneous boundary conditions. For that purpose, let us consider a secondsequence of subspaces Sµ ⊂ H1(Ω), Sµ ⊂ Sµ, which are not required to satisfy anyboundary conditions.
• Assumption 2, approximability: for any u ∈ Hm+1(Ω), any 0 ≤ i ≤ j ≤m + 1, and any µ ∈ N, there exists uI ∈ Sµ such that |u − uI |Hi(Ω) ≤Chj−i
µ ‖u‖Hj(Ω). If u = 0 on ∂Ω, then we can take uI ∈ Sµ.These two assumptions are formulated in more detail in Section 1.Our paper is divided into two parts. In the first part, we prove some general
approximation results for the Poisson problem with Dirichlet boundary conditions.The approximations (or discrete solutions) belong to some abstract spaces Sµ (forzero boundary conditions) or Sµ (for general boundary conditions) that are re-quired to satisfy certain reasonable assumptions (Assumptions 1 and 2, for zeroDirichlet boundary conditions, and Assumptions 1, 2, and 3 for non-zero boundary
4 I BABUSKA, V. NISTOR, AND N. TARFULEA
conditions). In the second part of the paper we provide examples of GeneralizedFinite Element Spaces satisfying these assumptions and extend the results of thefirst part to boundary data g with low regularity. Under Assumptions 1 and 2, ourmain approximation result in Section 1 is the following.
Theorem 0.1. Let Sµ ⊂ H1(Ω) be a sequence of finite dimensional subspacessatisfying Assumptions 1 and 2 for a sequence hµ → 0 and 1 ≤ p ≤ m. Letf ∈ Hp−1(Ω). Then the (unique) solutions u and uµ of Equations (1) and (4)satisfy
‖u− uµ‖H1(Ω) ≤ Chpµ‖u‖Hp+1(Ω) ≤ Chp
µ‖f‖Hp−1(Ω),
for constants independent of µ ∈ N and f ∈ Hp−1(Ω).
In Section 2 we extend our results to the non-homogeneous Dirichlet boundaryconditions case u = g on ∂Ω, with g ∈ Hm+1/2(∂Ω). In essence, we will be lookingfor a sequence Gk of approximate extensions of g, that is, a sequence of elements ofHm+1(Ω) subject to the following assumption. Recall that the sequence hµ shouldbe thought of as the “typical size” of the elements defining Sµ and satisfies hµ → 0.
• Assumption 3, approximate extensions: There exists a constant C > 0such that, for any g ∈ Hm+1/2(∂Ω), there exists a sequence Gk ∈ Sk
such that ‖Gk|∂Ω − g‖H1/2(∂Ω) ≤ Chmk ‖g‖Hm+1/2(∂Ω) and ‖Gk‖Hm+1(Ω) ≤
C‖g‖Hm+1/2(∂Ω).Let wk be the exact solution of −∆wk = f + ∆Gk in Ω, wk = 0 on ∂Ω. Also,
let (wk)µ ∈ Sµ be the discrete solution of this equation, namely, the solution of thediscrete variational problem
(3) B((wk)µ, v) = 〈f + ∆Gk, v〉L2(Ω), v ∈ Sµ,
where f ∈ Hm−1(Ω) is the data of Equation (14). The result we prove in Section 2is the following.
Theorem 0.2. Suppose Assumptions 1, 2, and 3 are satisfied and let us defineuk := (wk)k + Gk. Let 1 ≤ p ≤ m, f ∈ Hp−1(Ω), and g ∈ Hp+1/2(∂Ω). Thenthere exists a constant C > 0 such that the solution u ∈ Hp+1(Ω) of Equation (14)satisfies
‖u− uk‖H1(Ω) ≤ Chpk
(‖f‖Hp−1(Ω) + ‖g‖Hp+1/2(∂Ω)
).
In order to deal with low regularity boundary data, in Section 3 we considerthe Dirichlet problem −∆u = f in Ω, u = g on ∂Ω, with g ∈ H1/2−s(∂Ω) andf ∈ H−1−s(Ω), s > 0. Thus both f and g may be distributions. We say thatu = (u0, u1) ∈ H1−s(Ω) := H1−s(Ω) ⊕ H−1/2−s(∂Ω) is a weak solution of theproblem −∆u = f in Ω, u = g on the boundary ∂Ω if 〈u0,∆v〉Ω + 〈u1, v〉∂Ω =−〈f, v〉Ω + 〈g, ∂νv〉∂Ω, for all v ∈ H1+s(Ω). Then, the main result of Section 3 isthe following.
Theorem 0.3. Let g ∈ H1/2−s(∂Ω) and f ∈ H−1−s(Ω). Then there exists a uniqueweak solution u = (u0, u1) ∈ H1−s(Ω) for the problem −∆u = f in Ω, u = g onthe boundary ∂Ω and this solution satisfies
‖u0‖H1−s(Ω) + ‖u1‖H−1/2−s(∂Ω) ≤ CΩ,s
(‖g‖H1/2−s(∂Ω) + ‖f‖H−1−s(Ω)
),
for a constant CΩ,s that depends only on Ω and s.
DIRICHLET PROBLEM 5
The second part of this paper is dedicated to constructing concrete examples ofGeneralized Finite Element Spaces Sµ and Sµ satisfying the Assumptions 1, 2, and3 of the first part. In fact, we will prove that these assumptions are easy to fulfillwith a “flat-top” partition of unity and polynomial local approximation spaces. Theexact conditions are formulated in Section 4. The proof that the resulting GFEMspaces satisfy the Assumptions 1, 2, and 3 is in Section 5. In addition, in Section 6we also prove interior estimates for the error u − uk, where u is the solution ofthe distributional boundary value problem −∆u = 0 in Ω, u = g on ∂Ω, withdistributional data g ∈ H1/2−s(∂Ω), s > 0, and uk ∈ Sk is the discrete solution inour GFEM spaces. The last section contains some comments and a discussion ofsome further problems.
In this paper, we shall use the convention that C > 0 indicates a generic constant,independent of µ, which may be different each time when used, but is independentof the free variables of the formulas.
The second named author would like to acknowledge the generous support ofICES (Institute for Computational Sciences and Engineering) in Austin, Texas,while part of this paper was being written.
Part 1. Approximate Dirichlet boundary conditions
1. Homogeneous boundary conditions
In this section, we give a proof of Theorem 0.1. We begin by fixing the notationand then we prove some preliminary results.
Recall that Ω ⊂ Rn is a smooth, bounded domain, fixed throughout this paper.We shall fix in what follows m ∈ N = 1, 2, . . ., which will play the role of the orderof approximation. Also, f ∈ Hm−1(Ω) and u ∈ Hm+1(Ω)∩H1
0 (Ω) is the solution ofthe Poisson problem (1). We want to approximate u with functions uµ ∈ Sµ, µ ∈ N,where Sµ ⊂ H1(Ω) is a sequence of finite dimensional subspaces that satisfy theAssumption 1 and 2 formulated next. In those assumptions, the sequence hµ → 0should be thought of as the “typical size” of the elements defining Sµ. Our firstassumption is:
• Assumption 1 (nearly zero boundary values). There exists C > 0such that
‖vµ‖H1/2(∂Ω) ≤ Chmµ ‖vµ‖H1(Ω) for any vµ ∈ Sµ.
So Sµ does not necessarily consist of functions satisfying the Dirichlet bound-ary conditions. Let |u|H1(Ω) := [
∫Ω|∇u|2dx]1/2. Our second assumption will also
be used for non-homogeneous boundary conditions, so we formulate it also for asequence of spaces Sµ ⊂ H1(Ω), Sµ ⊂ Sµ.
• Assumption 2 (approximability): There exists C > 0 such that for any0 ≤ i ≤ j ≤ m+ 1, any u ∈ Hm+1(Ω), and any µ ∈ N, there exists uI ∈ Sµ
such that|u− uI |Hi(Ω) ≤ Chj−i
µ ‖u‖Hj(Ω).
If u = 0 on ∂Ω, then we can take uI ∈ Sµ.
We now proceed to the proof of Theorem 0.1. We first need some lemmas. Webegin with the following classical result [1, 16].
6 I BABUSKA, V. NISTOR, AND N. TARFULEA
Lemma 1.1. For v ∈ H1(Ω) there is a constant C that depends only on Ω suchthat
‖v‖2H1(Ω) ≤ C
[|v|2H1(Ω) + ‖v‖2
L2(∂Ω)
].
From this lemma we obtain that |vµ|H1(Ω) and ‖vµ‖H1(Ω) are equivalent normson Sµ, with equivalence bounds independent of µ.
Lemma 1.2. There exists C > 0 such that
C−1|vµ|H1(Ω) ≤ ‖vµ‖H1(Ω) ≤ C|vµ|H1(Ω)
for all µ large enough and all vµ ∈ Sµ.
Proof. From Lemma 1.1, we have
‖vµ‖2H1(Ω) ≤ C
[|vµ|2H1(Ω) + ‖vµ‖2
L2(∂Ω)
]≤ C|vµ|2H1(Ω) + Ch2m
µ ‖vµ‖2H1(Ω),
where the last inequality is a consequence of Assumption 1. Therefore, for µ large,hµ is small enough and we get
‖vµ‖H1(Ω) ≤ C(1− Ch2mµ )−1/2|vµ|H1(Ω)
which is enough to complete the proof.
Lemma 1.2 allows us now to introduce the discrete solution uµ of Equation (1)using the standard procedure. Let B(v, w) :=
∫Ω∇v · ∇wdx be the usual bilinear
form. For µ large, let us define the discrete solution uµ ∈ Sµ of the Poisson problem(1) by the usual formula
(4) B(uµ, vµ) =∫
Ω
f(x)vµ(x)dx, for all vµ ∈ Sµ.
Let ν be the outer unit normal to ∂Ω and dS denote the surface measure on ∂Ω.Similarly, let wµ ∈ Sµ, for µ large, be the solution of the variational problem
(5) B(wµ, vµ) =∫
∂Ω
∂νu(x)vµ(x)dS(x), for all vµ ∈ Sµ,
where u is the solution of Equation (1). Note that we need Lemma 1.2 to justifythe existence and uniqueness of the (weak) solutions uµ and wµ.
Lemma 1.3. Let u be the solution of the Poisson problem (1) and let uµ and wµ
be as in Equations (4) and (5). Then B(u−uµ−wµ, vµ) = 0 for all vµ ∈ Sµ; hence
|u− uµ − wµ|H1(Ω) ≤ |u− vµ|H1(Ω) for all vµ ∈ Sµ.
Proof. This is obtained from Assumption 1 as follows
(6) B(u, vµ) =∫
Ω
∇u · ∇vµdx =∫
Ω
fvµdx+∫
∂Ω
(∂νu)vµdS(x) = B(uµ +wµ, vµ),
for all vµ ∈ Sµ.
We now proceed to estimate uµ and wµ.
Lemma 1.4. Let u be the solution of the Poisson problem (1) and let wµ be thesolution of the weak problem (5). Then, for µ large, we have
(7) ‖wµ‖H1(Ω) ≤ Chmµ ‖u‖H2(Ω),
with C a constant independent of µ and u.
DIRICHLET PROBLEM 7
Proof. Since f ∈ L2(Ω), the solution u ∈ H2(Ω) and so, ∂νu is defined and belongsto H1/2(∂Ω). By the usual trace inequalities, we see that
where the last inequality follows from (8) and Assumption 1. Therefore ‖wµ‖H1(Ω) ≤Chm
µ ‖u‖H2(Ω), as claimed.
From this lemma we obtain the following.
Lemma 1.5. For µ large, the solution uµ of the weak problem (4) satisfies
(9) ‖uµ‖H1(Ω) ≤ C‖u‖H2(Ω),
with C a constant independent of µ and u.
Proof. Let us first observe that Lemmas 1.2 and 1.3 and Equation (8) give
‖uµ‖2H1(Ω) ≤ C|uµ|2H1(Ω) = CB(uµ, uµ) = C
[B(u, uµ)−B(wµ, uµ)
]= C
[B(u, uµ)− 〈∂νu, uµ〉L2(∂Ω)
]≤ C
[|B(u, uµ)|+ |〈∂νu, uµ〉L2(∂Ω)|
]≤ C‖u‖H1(Ω)‖uµ‖H1(Ω) + C‖∂νu‖L2(∂Ω)‖uµ‖L2(∂Ω)
≤ C‖u‖H2(Ω)‖uµ‖H1(Ω) + Chmµ ‖u‖H2(Ω)‖uµ‖H1(Ω).
Now it is easy to see that ‖uµ‖H1(Ω) ≤ C‖u‖H2(Ω), as claimed.
We are ready now to prove Theorem 0.1.
Proof. We shall assume p = m for simplicity; the proof in general is exactly thesame. Lemma 1.3 and the projection property, together with Lemma 1.4, give
(10) |u− uµ|H1(Ω) ≤ |u− uµ − wµ|H1(Ω) + |wµ|H1(Ω)
≤ |u− uI |H1(Ω) + Chmµ ‖u‖H2(Ω) ≤ Chm
µ ‖u‖Hm+1(Ω),
where for the last line we also used the approximation property (Assumption 2).The estimate in the H1-norm is obtained from Lemma 1.1, Equation (10), As-
sumption 1, and Lemma 1.4 as follows
‖u− uµ‖H1(Ω) ≤ C[|u− uµ|H1(Ω) + ‖uµ‖L2(∂Ω)
]≤ Chm
µ ‖u‖Hm+1(Ω) + Chmµ ‖uµ‖H1(Ω) ≤ Chm
µ ‖u‖Hm+1(Ω).
The proof is now complete.
In view of some further applications, we now include an error estimate in a“negative order” Sobolev norm. We let H−l(Ω) to be the dual of H l(Ω) with pivotL2(Ω). Since Ω is a smooth domain, H−l(Ω) can also be described as the closureof C∞(Ω) in the norm
(11) ‖u‖H−l(Ω) = supφ6=0
|〈u, φ〉L2(Ω)|‖φ‖Hl(Ω)
(Note that, in several other papers, H−l(Ω) denotes the dual of H l0(Ω).)
8 I BABUSKA, V. NISTOR, AND N. TARFULEA
Theorem 1.6. Let 0 ≤ l ≤ m, 1 ≤ p ≤ m, and γ = minl + p + 1,m. Then,under the assumptions of Theorem 0.1, the solutions u and uµ of Equation (1) andEquation (4), respectively, satisfy
‖u− uµ‖H−l(Ω) ≤ Chγµ‖u‖Hp+1(Ω) ≤ Chγ
µ‖f‖Hp−1(Ω),
for a constant C > 0 independent of µ and f ∈ Hp−1(Ω).
Proof. The proof of this theorem is an adaptation of the usual Nitsche-Aubintrick. Indeed, let us denote by F ∈ H l+2(Ω) the unique solution of the equation−∆F = φ, F = 0 on ∂Ω, for φ ∈ H l(Ω) arbitrary, non-zero. Then there existsa constant C > O, independent of φ, such that ‖F‖Hl+2(Ω) ≤ C‖φ‖Hl(Ω). ByAssumption 2, there exists FI ∈ Sµ such that
(12) |F − FI |H1(Ω) ≤ Chl+1µ ‖F‖Hl+2(Ω).
Then, the inequality (12) leads to the following easy observation, which will be usedlater,
In the following calculation, we shall use Equation (6) in the first inequality, andthen Theorem 0.1, Equations (12) and (13), 1.4, and 1.5 for the last inequality, toobtain
‖u− uµ‖H−l(Ω) = supφ6=0
|(u− uµ, φ)L2(Ω)|‖φ‖Hl(Ω)
= supφ6=0
∣∣B(u− uµ, F ) +∫
∂Ωuµ∂νFdS
∣∣‖φ‖Hl(Ω)
≤ supφ6=0
|B(u− uµ, F − FI)|‖φ‖Hl(Ω)
+ supφ6=0
|B(wµ, FI)|‖φ‖Hl(Ω)
+ supφ6=0
∣∣ ∫∂Ωuµ∂νFdS
∣∣‖φ‖Hl(Ω)
≤ supφ6=0
|u− uµ|H1(Ω)|F − FI |H1(Ω)
‖φ‖Hl(Ω)
+ supφ6=0
|wµ|H1(Ω)|FI |H1(Ω)
‖φ‖Hl(Ω)
+ supφ6=0
‖uµ‖L2(∂Ω)‖∂νF‖L2(∂Ω)
‖φ‖Hl(Ω)
≤ Chp+l+1µ ‖u‖Hp+1(Ω) + Chm
µ ‖u‖H2(Ω) + Chmµ ‖u‖H2(Ω)
≤ Chγµ‖u‖Hp+1(Ω),
by the definition of γ. This completes the proof.
2. Non-homogeneous boundary conditions
In this subsection we provide an approach to the non-homogeneous Dirichletboundary conditions. That is, we consider the boundary value problem
(14)
−∆u = f on Ω,u = g on ∂Ω.
Our approach is to reduce it to the case g = 0 and then to use the results on thePoisson problem (1). In a purely theoretical framework, this is achieved using anextension G of g and then solving the problem −∆w = f + ∆G, w = 0 on ∂Ω.The solution of (14) will then be u = w+G. This gives that the problem (14) hasa unique solution u ∈ Hp+1(Ω) for any f ∈ Hp−1(Ω) and g ∈ H1/2+p(∂Ω) and itsatisfies
‖u‖Hp+1(Ω) ≤ C(‖f‖Hp−1(Ω) + ‖g‖H1/2+p(∂Ω)
),
DIRICHLET PROBLEM 9
for a constant C > 0 that depends only on Ω and p ∈ Z+. (This result is valid alsofor p = 0.)
In practice, however, we need to slightly modify this approach since it is notpractical to construct the extension G (this is especially a problem if g has lowregularity, that is, if g is a distribution, for instance). We will be looking thereforefor a sequence Gk of approximate extensions of g, that is, satisfying the followingassumption. Recall that the sequence hµ should be thought of as the “typical size”of the elements defining Sµ and satisfies hµ → 0.
Assumption 3 (approximate extensions). We assume that there exist asequence of spaces Sk, Sk ⊂ Sk, satisfying Assumption 2 and a constant C > 0such that, for any g ∈ Hm+1/2(∂Ω), there exists a sequence Gk ∈ Sk such that‖Gk|∂Ω − g‖H1/2(∂Ω) ≤ Chm
k ‖g‖Hm+1/2(∂Ω) and ‖Gk‖Hm+1(Ω) ≤ C‖g‖Hm+1/2(∂Ω).
The proof of Theorem 0.2 follows below.
Proof. We can assume p = m. Remember that wk was introduced as the exactsolution to the boundary value problem −∆wk = f + ∆Gk in Ω, wk = 0 on ∂Ω.Let (wk)µ ∈ Sµ be the approximate solution of this equation, as in Equation (3).
We have that vk := wk +Gk solves the boundary value problem
−∆vk = f on Ω, vk = Gk on ∂Ω.
Hence the difference u − vk solve the boundary value problem ∆(u − vk) = 0,(u− vk) = g −Gk on ∂Ω. From this and Assumption 3 we obtain
From Equations (15) and (16) we obtain ‖u − uk‖H1(Ω) ≤ Chmk
(‖f‖Hm−1(Ω) +
‖g‖Hm+1/2(∂Ω)
), which is what we had to prove.
3. Distributional boundary data and the “inf-sup” condition
Let us consider the Dirichlet problem (14) (i.e., −∆u = f in Ω and u = g on∂Ω) with g ∈ H1/2−s(∂Ω) and f ∈ H−1−s(Ω), s ∈ R. If s ≤ 0, it is well known thatthe boundary value problem (14) has a unique solution u ∈ H1−s(Ω). Moreover,there is a constant CΩ,s, depending only on Ω and s ≤ 0, such that
‖u‖H1−s(Ω) ≤ CΩ,s(‖f‖H−1−s(Ω) + ‖g‖H1/2−s(∂Ω)).
In this section, we extend the above result to the case when g ∈ H1/2−s(∂Ω),with s > 0. Our approach is based on the so called “inf-sup” condition [3].
10 I BABUSKA, V. NISTOR, AND N. TARFULEA
3.1. The weak formulation. Let us define the functional space H1−s(Ω) :=H1−s(Ω) ⊕ H−1/2−s(∂Ω). Intuitively, for an element u = (u0, u1) ∈ H1−s(Ω),the first component u0 should be thought of as u in the interior of Ω, whilethe second component u1 represents the normal derivative ∂νu on ∂Ω. Let B :H1−s(Ω)×H1+s(Ω) → C be the bilinear functional defined by
B(u, v) := 〈u0,∆v〉Ω + 〈u1, v〉∂Ω,
where u = (u0, u1) ∈ H1−s(Ω).
Definition 3.1. Let g ∈ H1/2−s(∂Ω) and f ∈ H−1−s(Ω), s ∈ R. We say thatu = (u0, u1) ∈ H1−s(Ω) satisfies (14) in weak sense, or that u is a weak solution ofthe Dirichlet problem (14), if
(17) B(u, v) = −〈f, v〉Ω + 〈g, ∂νv〉∂Ω,
for all v ∈ H1+s(Ω).
Remark 3.2. If u = (u0, u1) ∈ H1−s(Ω) is a weak solution of the Dirichlet problem(14) in the sense of the above definition for s ≤ 0, then it is easy to see that thefirst component u0 is a classical solution for (14) and u1 = ∂νu0 on ∂Ω.
Remark 3.3. If u = (u0, u1) ∈ H1−s(Ω) is a solution of (14) in weak sense, then itis unique with this property.
The main ingredient we will use for proving Theorem 0.3 is the following “inf–sup” lemma (or the Babuska–Brezzi condition) [2, 3]. (This result was used forsimilar purposes in [9] in order to deal with low regularity Neumann data.)
Theorem 3.4. Let X and Y be reflexive Banach spaces with norms ‖ · ‖X and‖ · ‖Y . Also, let B1 : X × Y → C be a bilinear form. Assume that
(a) B1 is continuous;(b) There exists γ > 0 such that inf‖x‖=1 sup‖y‖≤1 |B1(x, y)| ≥ γ;(c) sup‖x‖X≤1 |B1(x, y)| > 0 whenever y 6= 0.
Then for any continuous functional F : Y → C there exists a unique x ∈ X suchthat F (y) = B1(x, y), for all y ∈ Y . Moreover, we have ‖x‖ ≤ ‖F‖/γ.
We are ready now to prove the main result of this section, that is, Theorem 0.3.
Proof. It is easy to see that B is continuous from its definition and the definition ofnegative order Sobolev spaces. Therefore, Condition (a) in Theorem 3.4 is satisfied.Let u = (u0, u1) ∈ H1−s(Ω) be such that ‖u‖ := ‖u0‖H1−s(Ω)+‖u1‖H−1/2−s(∂Ω) = 1.Since u also belongs to (H−1+s(Ω)⊕H1/2+s(∂Ω))∗, which is the dual of H1−s(Ω),there exists (v, v1) ∈ H−1+s(Ω) ⊕ H1/2+s(∂Ω) with ‖(v, v1)‖ := ‖v‖H−1+s(Ω) +‖v1‖H1/2+s(∂Ω) = 1 such that
(18) 〈u, (v, v1)〉 := 〈u0, v〉Ω + 〈u1, v1〉∂Ω ≥ 1/2.
Let V ∈ H1+s(Ω) ∩ H1/2+s(∂Ω) be the unique solution of the inhomogeneousDirichlet problem
(19)
∆V = v in Ω,V = v1 on ∂Ω.
Then,
(20) B(u, V ) = 〈u0,∆V 〉Ω + 〈u1, V 〉∂Ω = 〈u0, v〉Ω + 〈u1, v1〉∂Ω ≥ 1/2,
DIRICHLET PROBLEM 11
and this inequality implies that Condition (b) in Theorem 3.4 is also satisfied.Finaly, let us check Condition (c) in Theorem 3.4. Let v ∈ H1+s(Ω) such that
B(u, v) = 0, for all u = (u0, u1) ∈ H1−s(Ω). Then, v must satisfy the Dirichletproblem
(21)
∆v = 0 in Ω,v = 0 on ∂Ω,
whose unique solution is v = 0. This shows that Condition (c) in Theorem 3.4 issatisfied as well. The conclusion of the Theorem 0.3 follows if we take the functionalF in Theorem 3.4 to be F (v) := −〈f, v〉Ω + 〈g, ∂νv〉∂Ω.
Part 2. GFEM Approximation Spaces
4. The Generalized Finite Element Method
Our goal is to construct a sequence Sµ, µ = 1, 2, . . ., of Generalized FiniteElement spaces that satisfy the two assumptions of the previous section. To thisend, we shall introduce a sequence of Generalized Finite Element spaces that satisfycertain conditions (Conditions A(hµ), B, C, and D). In the following sections weshall prove that these conditions imply Assumptions 1 and 2.
We begin by recalling a few basic facts about the Generalized Finite ElementMethod [4, 8, 27]. This method is quite convenient when one needs test or trialspaces with high regularity.
4.1. Basic facts. Let k ∈ Z+. We shall denote as usual
|u|W k,∞(Ω) := max|α|=k
‖∂αu‖L∞(Ω), ‖u‖W k,∞(Ω) := max|α|≤k
‖∂αu‖L∞(Ω),
W k,∞(Ω) := u, ‖u‖W k,∞(Ω) < ∞, and ‖∇ω‖W k,∞(Ω) :=∑
We shall need the following slight generalization of a definition from [8, 27]:
Definition 4.1. Let Ω ⊂ Rn be an open set and ωjNj=1 be an open cover of Ω
such that any x ∈ Ω belongs to at most κ of the sets ωj . Also, let φj be a partitionof unity consisting of Wm,∞(Ω) functions and subordinated to the covering ωj(i.e., suppφj ⊂ ωj). If
(22) ‖∂αφj‖L∞(Ω) ≤ Ck/(diamωj)k, k = |α| ≤ m,
for any j = 1, . . . , N , then φj is called a (κ,C0, C1, . . . , Cm) partition of unity.
Assume also that we are given linear subspaces Ψj ⊂ Hm(ωj), j = 1, 2, . . . , N .The spaces Ψj will be called local approximation spaces and will be used to definethe space
(23) S = SGFEM := N∑
j=1
φjvj , vj ∈ Ψj
⊂ Hm(Ω),
which will be called the GFEM–space. The set ωj , φj ,Ψj will be called the set ofdata defining the GFEM–space S. A basic approximation property of the GFEM–spaces is the following Theorem from [8].
12 I BABUSKA, V. NISTOR, AND N. TARFULEA
Theorem 4.2 (Babuska-Melenk). We shall use the notations and definitions ofDefinition 4.1 and after. Let φj be a (κ,C0, C1) partition of unity. Also, letvj ∈ Ψj ⊂ H1(ωj), uap :=
∑j φjvj ∈ S, and dj = diamωj, the diameter of ωj.
Then
(24)
‖u− uap‖2L2(Ω) ≤ κC2
0
∑j
‖u− vj‖2L2(ωj)
and
‖∇(u− uap)‖2L2(Ω) ≤ 2κ
∑j
( C21‖u− vj‖2
L2(ωj)
(dj)2+ C2
0‖∇(u− vj)‖2L2(ωj)
).
4.2. Conditions on GFEM data defining Sµ. Recall that ω is star-shaped withrespect to ω∗ ⊂ ω if, for every x ∈ ω and every y ∈ ω∗, the segment with endpoints x and y is completely contained in ω. Also, recall that we have fixed aninteger m that plays the role of the order of approximation. Let ωj , φj ,ΨjN
j=1
be a single, fixed data defining a GFEM–space S, as in the previous subsection,and let Σ := ωj , φj ,Ψj , ω
∗j , where ωj is star-shaped with respect to ω∗j ⊂ ωj . We
shall need, in fact, to consider a sequence of such data
(25) Σµ = ωµj , φ
µj ,Ψ
µj , ω
∗µj Nµ
j=1, µ ∈ N,
defining GFEM–spaces Sµ
(26) Sµ := Nµ∑
j=1
φµj vj , vj ∈ Ψµ
j
⊂ Hm(Ω),
such that there exist constants A, Cj , σ, and κ and a sequence hµ → 0, as µ→∞,for which Σµ satisfies Conditions A(hµ), B, C, and D below for µ ∈ N. The sequencehµ gives the “typical size” of the elements defining Sµ, as in the first part.
Condition A(hµ). We have that Ω = ∪Nµ
j=1ωµj and for each j = 1, 2, . . . , Nµ, the
set ωµj is open of diameter dµ
j ≤ hµ ≤ 1 and ω∗µj ⊂ ωµj is an open ball of diameter
≥ σdµj such that ωµ
j is star-shaped with respect to ω∗µj .
Notice that we only assume the open covering ωµj to be nondegenerate, a
weaker condition than quasi-uniformity (see [16], Section 4.4, for definitions andmore information on these notions).
Condition B. The family φµj
Nµ
j=1 is a (κ,C0, C1, . . . , Cm) partition of unity.
The following condition defines the local approximation spaces Ψµj . To formulate
this condition, let us choose xj ∈ ωµj ∩ ∂Ω, if the intersection is not empty. We
can assume that linear coordinates have been chosen such that xj = 0 and thetangent space to ∂Ω at xj is xn = 0 = Rn−1. For hµ small, we can assumethat ωµ
j ∩ ∂Ω is contained in the graph of a smooth function gµj : Rn−1 → R. If
x = (x1, x2, . . . , xn) ∈ Rn, then we shall denote x′ = (x1, x2, . . . , xn−1) ∈ Rn−1, sothat x = (x′, xn). Let qµ
j : Rn−1 → R be a polynomial of order m such that(27)|∂αgµ
j (x′)− ∂αqµj (x′)| ≤ C(dµ
j )m+1−|α| for all (x′, xn) ∈ ωµj and |α| ≤ m+ 1.
This condition is satisfied, for instance, if ∂αgµj (0) = ∂αqµ
j (0), for all |α| ≤ m. Inthis case, the m-degree polynomial qµ
j : Rn−1 → R is uniquely defined by the aforementioned requirement.
DIRICHLET PROBLEM 13
Next, denote by qµj : Rn → Rn the bijective map
(28) qµj (x) = qµ
j (x′, xn) = (x′, xn + qµj (x′)).
Let us denote by Pk the space of polynomials of order at most k in n variables.
Condition C. We have Ψµj = Pm if ωµ
j ∩ ∂Ω = ∅ and, otherwise,
Ψµj = p (qµ
j )−1, p ∈ Pm, such that p(x′, 0) = 0,
where qµj are polynomials satisfying Equation (27) with a constant C independent
of j and µ.
An equivalent form of the condition “p ∈ Pm, p(x′, 0) = 0” is “p = xnp1,p1 ∈ Pm−1,” because any polynomial vanishing on the hyperplane xn = 0 isa multiple of xn. Since (qµ
Condition D. We have φµj = 1 on ω∗µj for all j = 1, . . . , Nµ for which ωµ
j ∩∂Ω 6= ∅.
The constants Cj , σ, and κ will be called structural constants. Note that wemust have Nµ →∞ as µ→∞.
The above assumptions are slightly weaker than the ones introduced in [9]. Forinstance, Condition C implies the following propety (which is similar to ConditionC in [9])
For any w ∈ Ψµj , any 0 ≤ l ≤ m+ 1, and any ball ω∗ ⊂ ωµ
j of diameter ≥ σdµj .
(29) ‖w‖Hl(ωµj ) ≤ C‖w‖Hl(ω∗).
For further applications, we shall also need a variant of the spaces Sµ in which noboundary conditions are imposed. Recall the functions qµ
j used to define the spacesΨµ
j . Let Ψµj = Ψµ
j if ωj does not touch the boundary ∂Ω and Ψµj = p (qµ
j )−1, p ∈Pm otherwise (the difference is that we no longer require p to vanish when xn = 0).We then define
(30) Sµ := Nµ∑
j=1
φµj vj , vj ∈ Ψµ
j
⊂ Hm(Ω).
We shall also need the following standard lemma, a proof of which, for s ∈ Z+,can be found in [9]. For s ≥ 0 it is proved by interpolation.
Lemma 4.3. Let ψj be measurable functions defined on an open set W and s ≥ 0.Assume that there exists an integer κ such that a point x ∈ W can belong to nomore than κ of the sets supp(ψj). Let f =
∑j ψj. Then there exists a constant
C > 0, depending only on κ, such that ‖f‖2Hs(W ) ≤ C
∑j ‖ψj‖2
Hs(W ).
Recall that dµj denotes the diameter of ωµ
j . Let us observe that Condition A(hµ)implies the following inverse inequality.
Lemma 4.4. There exists C > 0, depending only on σ, such that
(31) ‖p‖Hs(ωµj ) ≤ C(dµ
j )r−s‖p‖Hr(ω∗µj ),
for all 0 ≤ r ≤ s ≤ m, all j, all µ, and all polynomials p of order m.
14 I BABUSKA, V. NISTOR, AND N. TARFULEA
Proof. The proof of this lemma is inspired from the proof of (4.5.3) Lemma of [16].Since the diameters dµ
j are bounded uniformly in j and µ, it is enough to provethat
(32) |p|Hs(ωµj ) ≤ C(dµ
j )r−s|p|Hr(ω∗µj ),
for 0 ≤ r ≤ s ≤ m, all j, all µ, and all polynomials of order m.Consider µ and 1 ≤ j ≤ Nµ arbitrary, but fixed for the moment. Let
ωµj := 1
dµj
(x− xµj ), x ∈ ωµ
j , ω∗µj := 1dµ
j
(x− xµj ), x ∈ ω∗µj ,
where xµj is the center of the ball ω∗µj .
If p ∈ Pm is a polynomial of order m, then p is defined by p(x) := p(dµj x + xµ
j )for all x. Observe that the set Pm := p : p ∈ Pm is nothing but the set of allm-degree polynomials in x. Clearly,
(33) |p|Hk(ωµj ) = (dµ
j )k−n/2|p|Hk(ωµj ), for 0 ≤ k ≤ m.
We first prove (31) for the case r = 0. Since Pm is finite dimensional, we haveby the equivalence of norms on the unit ball B(0, 1) that
(34) ‖p‖Hk(B(0,1)) ≤ C‖p‖L2(B(0,1)), for any 0 ≤ k ≤ m,
where C > 0 is a constant that does not depend on k, j, and µ. From ConditionA(hµ), we obtain that
(35) ‖p‖L2(B(0,1)) ≤ C‖p‖L2(ω∗µj )
where C > 0 depends only on the structural constant σ. From (34) and (35), it isclear that
(36) ‖p‖Hk(ωµj ) ≤ C‖p‖L2(ω∗µ
j ) ∀p ∈ Pm,
where C > 0 depends only on σ. Therefore, (33) implies
|p|Hk(ωµj )(d
µj )k−n/2 ≤ C‖p‖L2(ω∗µ
j )(dµj )−n/2 for 0 ≤ k ≤ s,
from which we deduce that
|p|Hk(ωµj ) ≤ C(dµ
j )−k‖p‖L2(ω∗µj ) for 0 ≤ k ≤ s.
Since dµj ≤ hµ ≤ 1, we have
(37) ‖p‖Hs(ωµj ) ≤ C(dµ
j )−s‖p‖L2(ω∗µj ),
which is just (31) for r = 0.Let us now analyse the general case 0 ≤ r ≤ s ≤ m. For |α| = k, with s − r ≤
k ≤ s, Dαp = DβDγp for |β| = s− r and |γ| = k + r − s. Therefore,
‖Dαp‖L2(ωµj ) ≤ ‖Dγp‖Hs−r(ωµ
j )
≤ C(dµj )r−s‖Dγp‖L2(ω∗µ
j ) ( by (37))
≤ C(dµj )r−s|p|Hk+r−s(ω∗µ
j ).
Since|p|Hk(ωµ
j ) :=∑|α|=k
‖Dαp‖L2(ωµj ),
DIRICHLET PROBLEM 15
we obtain that
|p|Hk(ωµj ) ≤ C(dµ
j )r−s|p|Hk+r−s(ω∗µj ) for s− r ≤ k ≤ s.
Setting k = s in the above inequality gives Equation (32), and hence our desiredresult.
5. Properties of the spaces Sµ
In this section, we establish some properties of the GFEM spaces Sµ, µ ∈ N,defined in Equation (26) using the data Σµ = ωµ
j , φµj ,Ψ
µj , ω
∗µj Nµ
j=1 satisfying con-ditions A(hµ), B, C, and D introduced in the previous section for hµ → 0. Themain result is that the sequence Sµ satisfies Assumptions 1 and 2 of the first sec-tion. Also, we prove that there is a sequence Gk ∈ Sk of approximate extensionsof g which satisfies Assumption 3 in the case of the non-homogeneous Dirichletboundary-value problem (14).
Hereafter, for simplicity, we will omit the index µ whenever its appearance isimplicit.
Let us fix j such that ωj∩∂Ω is not empty. Recall the functions gj , qj : Rn−1 → Rdefined in the previous section. So, for h small, ωj∩∂Ω is contained in (x′, gj(x′)),the graph of the smooth function gj : Rn−1 → R (this may require a preliminaryrotation, which is not included in the notation, however, for the sake of simplicity).Let qj : Rn → Rn be the bijective map defined by Equation (28). Similarly, let
(38) gj(x) = gj(x′, xn) = (x′, xn + gj(x′)).
Then gj maps Rn−1 to a surface containing ωj ∩ ∂Ω. We have g−1j (x) = (x′, xn −
gj(x′)) and q−1j (x) = (x′, xn − qj(x′)).
We shall need the following estimate.
Lemma 5.1. For any polynomial p of order m, we have
Equation (40) then follows from Equation (39) and from the estimates qj(x′) =gj(x′) + O(dm+1
j ), ∂kqj(x′) = ∂kgj(x′) + O(dmj ) and |gj(x′)| ≤ dj for (x′, xn) ∈ ωj
(see Equation (27) and Condition C).By repeating the above arguments, the conclusion of the lemma follows induc-
tivelly for any s ≤ m+ 1.
Remark 5.2. Let us observe that Condition A(hµ) was used implicitly in the proofof Lemma 5.1 when we used the inverse estimates dk−1
j ‖p‖Hk(ωj) ≤ C‖p‖H1(ωj).
Corollary 5.3. Let p ∈ Pm, then
‖φj
(p g−1
j − p q−1j
)‖H1(ωj) ≤ Cdm
j ‖p‖H1(ωj).
If p ∈ Pm also vanishes on xn = 0 then we have
‖φj(p q−1j )‖L2(∂Ω) ≤ Cdm
j ‖p‖H1(ω∗j ).
Here C is a constant independent of p, µ, and j.
Proof. Using Lemma 5.1 and Assumption B, we obtain
‖φj
(p g−1
j − p q−1j
)‖H1(ωj) ≤ ‖φj‖L∞(ωj)‖p g
−1j − p q−1
j ‖H1(ωj)
+(Σn
i=1‖∂iφj‖L∞(ωj)
)‖p g−1
j − p q−1j ‖L2(ωj)
≤ Cdmj ‖p‖H1(ωj) + Cd−1
j dm+1j ‖p‖H1(ωj) ≤ Cdm
j ‖p‖H1(ωj).
The last part follows from the first part of this corollary, which we have alreadyproved, and from the fact that φj(p g−1
j ) = 0 on ∂Ω. Indeed,
‖φj(p q−1j )‖L2(∂Ω) = ‖φj
(p g−1
j − p q−1j
)‖L2(∂Ω)
≤ C‖φj
(p g−1
j − p q−1j
)‖H1(Ω) = C‖φj
(p g−1
j − p q−1j
)‖H1(ωj)
≤ Cdmj ‖p‖H1(ωj) ≤ Cdm
j ‖p‖H1(ω∗j )
The proof is now complete.
We are ready now to prove that Assumption 1 is satisfied by the sequence ofGFEM–spaces Sµ introduced in Subsection 4.2.
Proposition 5.4. Let Sµ be the sequence of GFEM–spaces defined by data Σµ
(Equation (25)) satisfying conditions A(hµ), B, C, and D. Then the sequence Sµ
satisfies Assumption 1.
18 I BABUSKA, V. NISTOR, AND N. TARFULEA
Proof. Let wj ∈ Ψµj and w =
∑φjwj ∈ Sµ. Since we are interested in evaluating w
at ∂Ω, we can assume that only the terms corresponding to j for which ωj ∩∂Ω 6= ∅appear in the sum. Then wj = pj q−1
j , for some polynomials pj ∈ Pm vanishingon xn = 0. Hence Lemma 4.3 and Corollary 5.3 give
‖w‖2H1/2(∂Ω) ≤ C
∑j
‖φjwj‖2L2(∂Ω) = C
∑j
‖φj(pj q−1j )‖2
L2(∂Ω)
≤ C∑
j
d2mj ‖pj‖2
H1(ω∗j ) ≤ Ch2mµ
∑j
‖pj‖2H1(ω∗j ) ≤ Ch2m
µ
∑j
‖wj‖2H1(ω∗j ).
By Condition D,∑
j ‖wj‖2H1(ω∗j ) = ‖w‖2
H1(∪ω∗j ). Therefore,
‖w‖2L2(∂Ω) ≤ Ch2m
µ ‖w‖2H1(∪ω∗j ) ≤ Ch2m
µ ‖w‖2H1(Ω).
Assumption 1 is hence satisfied by taking square roots.
Remark 5.5. Condition D is only needed in the proof of Proposition 5.4. Althoughone can prove that
(41)∑
j
‖wj‖2H1(ω∗j ) ≤ C‖w‖2
H1(Ω)
(by using norm equivalence in finite dimensional spaces), one can not bypass Con-dition D because the constant C in (41) depends on µ. To remove this dependence,one would have to impose additional and/or different conditions on the partitionof unity.
The proof that the sequence Sµ also satisfies Assumption 2 is also based onthe above lemma and on the following result. Recall that the local approximationspaces Ψj and Ψµ
j were defined in Subsection 4.2.
Lemma 5.6. Let u ∈ Hm+1(ωj). Then there exists a polynomial w ∈ Ψµj such that
‖u−w‖Hi(ωj) ≤ Cds−ij ‖u‖Hs(ωj), 0 ≤ i ≤ s ≤ m+ 1, for a constant C independent
of u, µ, and j. If u = 0 on ωj ∩ ∂Ω, then we can choose w ∈ Ψµj .
Proof. We are especially interested in the case when u = 0 on ωj ∩ ∂Ω, so we shalldeal with this case in detail. The other one is proved in exactly the same way.
Let us consider v = u gj . Since gj maps Rn−1 = xn = 0 to a surfacecontaining ωj ∩ ∂Ω, we obtain that v = 0 on Rn−1. For hµ small enough, we canassume that g−1
j (ωj) lies on one side of Rn−1. Let U be the union of the closure ofg−1
j (ωj) and of its symmetric subset with respect to Rn−1. Define v1 ∈ H1(U) tobe the odd extension of v (odd with respect to the reflection about the subspaceRn−1). Let p1 be the projection of v1 onto the subspace Pm of polynomials ofdegree m on U . This projection maps even functions to even functions and oddfunctions to odd functions. Hence p1 is also odd. In particular, p1 = 0 on Rn−1.We also know from standard approximation results [16] that
‖v1 − p1‖Hi(U) ≤ Cds−ij ‖v1‖Hs(U).
Then
‖u− p1 g−1j ‖Hi(ωj) ≤ C‖v1 − p1‖Hi(U) ≤ Cds−i
j ‖v1‖Hs(U) ≤ Cds−ij ‖u‖Hs(ωj).
DIRICHLET PROBLEM 19
Let w = p1 q−1j . The lemma follows from
‖u− w‖Hi(ωj) ≤ ‖u− p1 g−1j ‖Hi(ωj) + ‖p1 g−1
j − p1 q−1j ‖Hi(ωj)
≤ Cds−ij ‖u‖Hm+1(ωj) + Cdm+1−i
j ‖p1‖H1(ωj) ≤ Cds−ij ‖u‖Hs(ωj),
where we have also used Lemma 5.1.
We are ready now to prove Assumption 2. See [4], section 6.1, and [9] for relatedresults.
Proposition 5.7. The sequence of GFEM spaces Sµ satisfies Assumption 2.
Proof. We proceed as in [9], Theorem 3.2. Let u ∈ Hm+1(Ω). If ωl does notintersect ∂Ω, we define wl ∈ Ψl = Pm to be the orthogonal projection of u onto Pm
in H1(ωl). Otherwise, we define wl ∈ Ψl using Lemma 5.6. Then let w =∑
l φlwl.By using Lemma 5.6, the definition of the local approximation spaces Ψl (ConditionC), and the bounds on |φl|W k,∞(Ω) (Condition B), we obtain
|u− w|2Hi(Ω) =∣∣ ∑
l
φl(u− wl)∣∣2Hi(Ω)
≤ C∑
l
|φl(u− wl)|2Hi(ωl)
≤ C∑
l
i∑s=1
|φl|2W s,∞(ωl)|u− wl|2Hi−s(ωl)
≤ C∑
l
i∑s=1
Csd−2sl |u− wl|2Hi−s(ωl)
≤ C∑
l
i∑s=1
Csd−2sl d
2(j−i+s)l ‖u‖2
Hj(ωl)
≤ Ch2(j−i)µ ‖u‖2
Hj(Ω).
This completes the result.
Next, we will be looking for a sequence Gk of approximate extensions of g in Sk.Recall that the spaces Sk ⊃ Sk were defined in Equation (30) and are variants ofthe spaces Sk that are not required to satisfy, even approximately, the boundaryconditions. The construction of such a sequence Gk of approximate extension aswell as the analysis of the resulting method are the main results of this subsection.Other methods for constructing Gk are certainly possible.
We now check that it is possible to choose Gk ∈ Sk satisfying Assumption 3. Wefollow the method in [3].
Proposition 5.8. There exist continuous linear maps Ik : Hm+1(Ω) → Sk, suchthat
(42) |u− Ik(u)|Hr(Ω) ≤ Chm+1−rk ‖u‖Hm+1(Ω),
for r = 0, 1, . . . ,m+ 1.
20 I BABUSKA, V. NISTOR, AND N. TARFULEA
Proof. For u ∈ Hm+1(Ω) and j fixed, let v = u gj . The Taylor polynomial ofdegree m of v averaged over g−1(ωj) is given by
(43) Pj(x) := Qmj v(x) =
∫g−1(ωj)
Qy,v,n(x)Φj(y) dy,
where
Qy,v,n(x) = v(y)+n∑
i=1
∂iv(y)(xi−yi)+. . .+∑|α|=m
v(α)(y)α!
(x−y)α, α! = α1! . . . αn!,
is the Taylor polynomial of v at y of degree m and Φj ∈ C∞c (g−1(ωj)) is a function
with integral 1. Then, by the Bramble–Hilbert Lemma, we have
(44) |v − Pj |Hs(g−1(ωj)) ≤ Chm+1−sk |v|Hm+1(g−1(ωj)), for all 0 ≤ s ≤ m+ 1.
Consider wj := Pj q−1j ∈ Ψj . Let w :=
∑j φjwj . Then,
(45) |u− w|2Hr(Ω) ≤ C∑
j
|φj(u− w)|2Hr(Ω) ≤ C∑
j
|φj(u− w)|2Hr(ωj)
≤ C∑
j
r∑i=0
|φj |2W i,∞(ωj)|u− wj |2Hr−i(ωj)
≤ C∑
j
r∑i=0
|φj |2W i,∞(ωj)[|u− Pj g−1
j |2Hr−i(ωj)+ |Pj g−1
j − Pj q−1j |2Hr−i(ωj)
].
By changing variables and (44), we obtain
(46)|u− Pj g−1
j |2Hr−i(ωj)= |v g−1
j − Pj g−1j |2Hr−i(ωj)
≤ C|v − Pj |2Hr−i(g−1j (ωj))
≤ Ch2(m+1−r+i)k |v|2Hm+1(g−1(ωj))
= Ch2(m+1−r+i)k |u gj |2Hm+1(g−1(ωj))
≤ Ch2(m+1−r+i)k |u|2Hm+1(ωj)
.
Also, from Lemma 5.1 and the definition (43) of Pj , we have
(47) |Pj g−1j − Pj q−1
j |2Hr−i(ωj)≤ Ch
2(m+1−r+i)k ‖Pj‖2
H1(ωj)
≤ Ch2(m+1−r+i)k ‖u‖Hm+1(ωj).
From (45), (46), (47), and Condition B, it follows that
(48)
|u− w|2Hr(Ω) ≤ C∑
j
r∑i=0
h−2ik [h2(m+1−r+i)
k |u|2Hm+1(ωj)+ h
2(m+1−r+i)k ‖u‖2
H1(ωj)]
≤ Ch2(m+1−r)k
∑j
‖u‖2Hm+1(ωj)
≤ Cκh2(m+1−r)k ‖u‖2
Hm+1(Ω),
for all 0 ≤ r ≤ m+ 1.Define Ik(u) := w. Clearly Ik is a linear map from Hm+1(Ω) to Sk which satisfies
(42). This ends the proof of the proposition.
From this proposition we obtain right away the Assumption 3.
DIRICHLET PROBLEM 21
Proposition 5.9. For any g ∈ Hm+1/2(∂Ω) there exists a sequence Gk ∈ Sk
satisfying Assumption 3.
Proof. Indeed, let us chose G ∈ Hm+1(Ω) that extends g to the interior andsatisfies ‖G‖Hm+1(Ω) ≤ C‖g‖Hm+1/2(∂Ω), with C independent of g. Then chooseGk = Ik(G), with Ik as in Proposition 5.8.
6. Interior numerical approximation
In this section, we construct a sequence of approximations uk ∈ Sk of the solutionu of the distributional boundary value problem ∆u = 0 in Ω, u = g on ∂Ω, withg ∈ H1/2−s(∂Ω), and we prove interior estimates for the error u−uk. The sequenceof spaces Sk is the sequence of GFEM spaces constructed in Section 4 and hence itsatisfies the Assumptions 2 and 3, by the results of Section 5. In particular, Sk ⊂ Sk.We need to consider GFEM spaces instead of the more general framework of thefirst part because we need the interior approximation result of [9] recalled below.
In this section, s ∈ Z+ is fixed. Our results mirror the ones in [9], where theNeumann problem was considered. The approach is different however, in part be-cause it takes more work to construct finite element approximations of the solutionu in the case of the Dirichlet boundary conditions.
Our approach is to first approximate g with a sequence Gk of functions. Theneach of the approximate equations ∆vk = 0, vk = Gk at the boundary, is solvedapproximately using the results of the previous sections. This yields, for any k, asequence (vk)µ with (vk)µ ∈ Sµ. The desired sequence of approximations of thesolution u is then uµ := (vµ)µ ∈ Sµ. Our approach is thus similar to that of Section2.
We shall use the spaces Sµ and Sµ that appear in Assumptions 1, 2, and 3. Letus also recall that Sµ ⊂ Sµ. The definition of the space Sµ is slightly different fromthe one in [9]; however, the difference is only in the local approximation spacesat the boundary, and hence this does not affect the spaces S<
µ (Ω) := Sµ ∩ C∞c (Ω).Therefore, it follows that Theorem 3.12 from [9] is still valid, the proof being exactlythe same. Namely, we have the following:
Theorem 6.1. Let A b B ⊂ Ω be open subsets. Then there exists C > 0 with thefollowing property. If u ∈ H1(Ω) and uµ ∈ Sµ are such that B(u − uµ, χ) = 0 forall χ ∈ S<
µ (Ω), then for hµ small enough,
‖u− uµ‖H1(A) ≤ C(
infχ∈Sµ
‖u− χ‖H1(B) + ‖u− uµ‖H−m(B)
).
The constant C depends only on A, B, and the structural constants, but not onµ ∈ Z+.
This result is the version for the Generalized Finite Element of a basic result byNitsche and Schatz [31, Theorem 5.1]. See also [34, Theorem 9.2]. The above resultis the reason why we work in this section in the framework of the Generalized FiniteElement Method and not in the abstract setting of the first part of this paper.
Recall that s is fixed in this section. We otherwise use the notation of theprevious sections. We shall need the following property of the spaces Sj .
• The low regularity approximate extension property: There exists a constantC > 0 such that, for any g ∈ H1/2−s(∂Ω), we can find a sequence Gj ∈ Sj
22 I BABUSKA, V. NISTOR, AND N. TARFULEA
satisfying ‖Gj |∂Ω − g‖H1/2−s−γ(∂Ω) ≤ Chγj ‖g‖H1/2−s(∂Ω) for all 0 ≤ γ ≤ m
and ‖Gj‖H2(Ω) ≤ Ch−s−1j ‖g‖H1/2−s(∂Ω).
Lemma 6.2. Assumptions 2 and 3 imply the low regularity approximate exten-sion property. In particular, the spaces Sj satisfy the low regularity approximateextension property.
Proof. Let g ∈ H1/2−s(∂Ω). As in [32], we shall consider for each H > 0 functionsGH ∈ C∞(Ω) such that
‖g −GH‖H1/2−s−γ(∂Ω) ≤ CHγ‖g‖H1/2−s(∂Ω)
‖GH‖H2(Ω) ≤ CH−s−1‖g‖H1/2−s(∂Ω),
with constants independent of g. On Rn, we can chose GH = χH ∗g, with χH(x) =H−nχ(x/H) for a suitable chosen χ ∈ C∞c (Rn). In general, this procedure can belocalized as in [15].
Let us then choose H = hj and let Gj ∈ Sj be the H2-projection of GH ontoSj , namely
(GH −Gj , v)H2(Ω) :=∑|α|≤2
∫Ω
∂α(GH −Gj)∂αv dx = 0, v ∈ Sj .
Then ‖Gj‖H2(Ω) ≤ ‖GH‖H2(Ω) ≤ Ch−s−1j ‖g‖H1/2−s(∂Ω). To prove our result, it is
enough to show that the restrictions to the boundary ∂Ω satisfy
(49) ‖Gj −GH‖H1/2−l(∂Ω) ≤ Chl+1j ‖GH‖H2(Ω),
for any l ≥ 0 (for our result we then take l = s+ γ).We shall proceed as in [3]. Let w ∈ H l−1/2(∂Ω) arbitrary, but fixed for the mo-
ment. We letW ∈ H2(Ω) be the unique solution of the weak problem (W, f)H2(Ω) =(w, f)L2(∂Ω), for all f ∈ H2(Ω). (The solution W is obtained simply by using theRiesz representation of the functional f → (f, w)L2(∂Ω), which is continuous onH2(Ω) by the trace theorem for Sobolev spaces. We note that W satisfies theelliptic equation ∆2W − ∆W + W = 0, with suitable Neumann type boundaryconditions.) Nirenberg’s trick [1, 26, 28, 33], then shows that W ∈ H l+3(Ω) and‖W‖Hl+3(Ω) ≤ C‖w‖Hl−1/2(∂Ω). Hence Assumption 2 (approximation property)gives that there exists χ ∈ Sj such that
‖W − χ‖H2(Ω) ≤ Chl+1j ‖W‖Hl+3(Ω) ≤ Chl+1
j ‖w‖Hl−1/2(∂Ω).
We obtain
‖Gj −GH‖H1/2−l(∂Ω) = supw 6=0
|(Gj −GH , w)L2(∂Ω)|‖w‖Hl−1/2(∂Ω)
= supw 6=0
|(Gj −GH ,W )H2(Ω)|‖w‖Hl−1/2(∂Ω)
= supw 6=0
|(Gj −GH ,W − χ)H2(Ω)|‖w‖Hl−1/2(∂Ω)
≤ C‖Gj −GH‖H2(Ω) supw 6=0
‖W − χ‖H2(Ω)
‖w‖Hl−1/2(∂Ω)
≤ Chl+1j ‖Gj −GH‖H2(Ω) ≤ Chl+1
j ‖GH‖H2(Ω).
This proves Equation (49) and hence completes the proof of the Lemma.
We are now ready to prove a result on the interior approximation properties forboundary value problems with low regularity (i.e., distributional) boundary data.
DIRICHLET PROBLEM 23
Theorem 6.3. Let Sµ ⊂ Sµ be our sequences of GFEM spaces. Also, let g ∈H1/2−s(∂Ω), 1 ≤ s ≤ m−1, A b Ω, and Gj be as in the low regularity approximateextension property. We define (vj)µ to be the discrete solution of the equation∆u = 0, u = Gj at the boundary, as defined in Equation (55) and uµ := (vµ)µ.Then
‖u− uµ‖H1(A) ≤ Chm−s−1µ ‖g‖H1/2−s(∂Ω).
Proof. We now proceed as in Section 2. Let us denote by wk the solution of
(50) −∆wk = ∆Gk ∈ L2(Ω) in Ω and wk = 0 on ∂Ω,
Then we let vk := wk +Gk, which will satisfy ∆vk = 0 in Ω, vk = Gk on ∂Ω.Lemma 6.2 shows that the low regularity approximate extension property is
satisfied. Theorem 0.3 and the low regularity approximate extension property thengive
Let B be an open set such that A b B b Ω and t be a parameter. Then forany harmonic function φ ∈ C∞(Ω), there exists a constant C such that ‖φ‖Ht(B) ≤C‖φ‖H1−s−γ(Ω). Taking φ := u−vk, t = 1, and using also Equation (51), we obtain
In spite of all the differences in assumptions and definitions between [11, 30] andour paper, the main issue seems to be in all of these papers to provide simple exam-ples of spaces satisfying the various assumptions imposed in each of these paperson the approximation spaces. In particular, it would be interesting to provide otherexamples of spaces Sµ satisfying Assumptions 1 and 2. It would also be interestingto see if a modification of the uniform partition of unity can give, by restriction,spaces Sµ satisfying these Assumptions. A related problem is to construct otherexamples of spaces satisfying the interior estimates used in Section 6 as well as thegeneral Assumptions 1, 2, and 3. Then Theorem 6.3 would be valid for these spacesas well. Finally, it would be important to integrate our results with the issues aris-ing from numerical integration and to provide explicit numerical examples testingour results. Related numerical tests together with some theoretical results can befound, for example, in [5, 20, 32].
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University of Texas at Austin, Institute for Computational Engineering and Sci-
